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Assume the second term, \(4\sqrt{3}\), of the given area is a clue: it's likely to be the area of the equilateral triangle.Divide 48 by 3 (squares) = 16. Square side = 4. Check: Length = 4 must equal equilateral triangle side. Area of equilateral triangle if side = 4 is \(\frac{4^2\sqrt{3}}{4} = 4\sqrt{3}\) - CorrectSquare side = 4, there are nine sides on the perimeter, 9 * 4 = 36Answer C

Longer version

Area generally: formulas and given value

Knowing the formula for the area of an equilateral triangle makes this problem a lot easier.*

Total area = (area of three squares) + (area of equilateral triangle)

Area of three square \((3)s^2\)

Area of equilateral triangle = \(\frac{s^2\sqrt{3}}{4}\)

Total area: \(3s^2 + \frac{s^2\sqrt{3}}{4}\)

Total area = \(48 + 4√3\)

Find side length of square from area of square set equal to 48

Gamble a little. The second given term, \(4\sqrt{3}\) is likely to be the area of the equilateral triangle.So assume that the integer portion of the given area will yield a square's side length

Set just the integer portion of the area, 48, equal to the area of the three squares

\(3s^2 = 48\)\(s^2 = 16\)\(s = 4\)

Check: square side = triangle side, find triangle area

Each square: has equal side lengths that are equal to the side of the equilateral triangle (each square shares a side with the triangle)If square side length = 4, triangle side length must = 4

Set the other given term equal to the area of the equilateral triangle\(\frac{s^2\sqrt{3}}{4} =4√3\)

\({s^2\sqrt{3}}=16√3\)

\(s^2 = 16\)\(s = 4\)

Correct: the side of square = side of triangle = 4

Perimeter

Perimeter = 9 square side lengths: \(9 * 4 = 36\)

Answer C

*If you don't remember the formula for the area of an equilateral triangle, draw one. Drop an altitude, which is a perpendicular bisector of the opposite side and of the vertex.That altitude creates two congruent right 30-60-90 triangles with side lengths that correspond to 30-60-90, in ratio \(x : x\sqrt{3} : 2x\)

Assume the second term, \(4\sqrt{3}\), of the given area is a clue: it's likely to be the area of the equilateral triangle.Divide 48 by 3 (squares) = 16. Square side = 4. Check: Length = 4 must equal equilateral triangle side. Area of equilateral triangle if side = 4 is \(\frac{4^2\sqrt{3}}{4} = 4\sqrt{3}\) - CorrectSquare side = 4, there are nine sides on the perimeter, 9 * 4 = 36Answer C